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.Thec2nC.(5.4)special role of position in classical physics can be tracedn2mn nnto the nature of interactions that depend on distance(Zurek, 1981, 1982, 1991) and therefore commute with The effect of the environment can be expressedposition [see Eq.(4.22)].Evolution of open systems in- through the propagator J acting on the reduced :Scludes, however, the flow in phase space induced by theself-Hamiltonian.Consequently a set of preferred statesx,x ,t dx0dx 0J x,x ,t x0 ,x 0 ,t0 x0 ,x 0 ,t0.S Sturns out to be a compromise, localized in both position(5.5)and momentum, localized in phase space.We focus on the case in which the system and the envi-Einselection is responsible for the classical structureronment are initially statistically independent, so thatof phase space.States selected by the predictability sievetheir density matrices start from a product state:become phase-space points, and their time-orderedsequences turn into trajectories.In underdamped, clas-.(5.6)SE S Esically regular systems one can recover this phase-spaceThis is a restrictive assumption.One can try to justify itstructure along with (almost) reversible evolution.Inas an idealization of a measurement that correlates Schaotic systems there is a price to be paid for classicality:with the observer and destroys correlations of S with E,combination of decoherence with the exponential diver-but that is only an approximation, since realistic mea-gence of classical trajectories (which is the defining fea-surements leave partial correlations with the environ-ture of chaos) leads to entropy production at a ratement intact.Fortunately, such preexisting correlationsgiven in the classical limit by the sum of positivelead to only minor differences in the salient features ofLyapunov exponents.Thus the second law of thermody-the subsequent evolution of the system (Anglin, Paz,namics can emerge from the interplay of classical dy-and Zurek, 1997; Romero and Paz, 1997).namics and quantum decoherence, with entropy produc-The evolution of the whole can be represented asSEtion caused by information leaking into theenvironment (Zurek and Paz, 1994, 1995a; Zurek,x,q,x ,q ,tSE1998b; Paz and Zurek, 2001).dx0dx 0dq0dq 0 x0 ,q0 ,x 0 ,q 0 ,t0SEA.Quantum Brownian motion*K x,q,t,x0 ,q0 K x ,q ,t,x 0 ,q 0.(5.7)The quantum Brownian motion model consists of anenvironment E a collection of harmonic oscillators (co-Above, we suppress the sum over the indices of the in-ordinates qn , masses mn , frequencies , and couplingndividual environment oscillators.The evolution operatorconstants cn) interacting with the system S (coordinateK(x,q,t,x0 ,q0) can be expressed as a path integralx), with a mass M and a potential V(x).We shall often2consider harmonic V(x) M x2/2 so that the whole iK x,q,t,x0 ,q0 DxDq exp I x,q , (5.8)SE is linear and one can obtain an exact solution.Thisassumption will be relaxed later.where I x,q is the action functional that depends onThe Lagrangian of the system-environment entity isthe trajectories x and q.The integration must satisfy theL x,qn LS x LSE x, qn ; (5.1)boundary conditionsthe system alone has the Lagrangianx 0 x0 ; x t x; q 0 q0 ; q t q.(5.9)M MThe expression for the propagator of the density matrix2LS x ‹ 2 V x ‹ 2 x2.(5.2)can now be written in terms of actions corresponding to2 2the two Lagrangians, Eqs.(5.1) (5.3):The effect of the environment is modeled by the sum ofthe Lagrangians of individual oscillators and of the J x,x ,t x0 ,x 0 ,t0system-environment interaction terms:i2DxDx exp IS x IS xmn 2 cnxÙLSE q2 qn.(5.3)n n2 mn 2nndqdq0dq 0 q0 ,q 0This Lagrangian takes into account the renormalizationEof the potential energy of the Brownian particle.Theiinteraction depends (linearly) on the position x of theDqDq exp ISE x,q ISE x ,q.harmonic oscillator.Hence we expect x to be an instan-htaneous pointer observable.In combination with the(5.10)harmonic evolution this leads to Gaussian pointer states,well localized in both x and p.An important character- The separability of the initial conditions, Eq.(5.6), wasistic of the model is the spectral density of the environ- used to make the propagator depend only on the initialment: conditions of the environment.Collecting all terms con-Rev.Mod.Phys., Vol.75, No.3, July 2003 738Wojciech Hubert Zurek: Decoherence, einselection, and the quantum origins of the classicaltaining integrals over E in the above expression leads to The time-dependent coefficients bk and aij are com-the influence functional (Feynman and Vernon, 1963) puted from the noise and dissipation kernels, which re-flect properties of the environment.They obtain fromthe solutions of the equationF x,x dqdq0dq 0 q0 ,q 0Es2i ü s u s 2 ds s s u s 0, (5.17)DqDq exp ISE x,q ISE x ,q.where is the bare frequency of the oscillator.Two(5.11)such solutions that satisfy the boundary conditionsInfluence functional can be evaluated explicitly foru1(0) u2(t) 1 and u1(t) u2(0) 0 can be used forspecific models of the initial density matrix of the envi-this purpose.They yield the coefficients of the Gaussianronment.An environment in thermal equilibrium pro-propagator throughvides a useful and tractable model for the initial state.Ù Ùb1 2 t u2 1 t /2, b3 4 t u2 1 0 /2, (5.18a)The density matrix of the nth mode of the thermal en-vironment ist t1aij t ds ds ui s uj s s s.1 ij 0 0mn nq,qEn(5.18b)n2 sinhThe master equation can now be obtained by takingkBTthe time derivative of Eq.(5.5), which in effect reducesmn nto the computation of the derivative of the propagator,expn Eq.(5.16), above:2 sinhkBTÙJ 3 /b3 i 1XY i 2X0Y i 3XY0 i 4X0Y0nÙ Ù Ùa11Y2 a12YY0 a22Y2 J.(5.19)q2 q n2 cosh 2qnq n.0nkBTThe time derivative of can be obtained by multiplyingSthe operator on the right-hand side by an initial density(5.12)matrix and integrating over the initial coordinatesX0 ,Y0.Given the form of Eq.(5.19), one may expectThe influence functional F can be written as (Grabert,that this procedure will yield an integro-differentialSchramm, and Ingold, 1988)(nonlocal in time) evolution operator for.However,St sthe time dependence of the evolution operator disap-i ln F x,x ds x x s du s s x x s0 0pears as a result of the two identities satisfied by thepropagator:i s s x x s , (5.13)b1 iwhere (s) and (s) are known as the dissipation andY0J Y J, (5.20a)b3 b3 Xnoise kernels, respectively, and are defined in terms ofthe spectral density:b1 i 2a11 a12b1X0J X i Yb2 b2 Y b2 b2b3s d C coth /2 cos s ; (5.14)a12J.(5.20b)b2b3 Xs d C sin s.(5.15)After the appropriate substitutions, the resulting equa-tion with renormalized Hamiltonian Hren has the formWith the assumption of thermal equilibrium at kBT1/ , and in the harmonic-oscillator case V(x) i2 Ùx,x ,t x Hren t , x t x xS SM x2/2, the integrand of Eq.(5.10) for the propaga-tor is Gaussian.The integral can be computed exactly2D t x x x,x ,tand should also have a Gaussian form.The result can be x x Sconveniently written in terms of the diagonal and off-if t x x x,x ,t.x x Sdiagonal coordinates of the density matrix in the posi-(5
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